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It sounds like you might have drained your battery! When the car is running does the Cigarette lighter produce power? It should be working while the car is running. That would also explain why the car starts sporadically. Try leaving the car on for 40 minutes. That might charge the battery enough.

Your Mazda definitely has a safety fuse controlling the cig. lighter...Review your manual for fuse block location...(sometimes more than one:) the fuse diagraham in the manual should help find the specific fuse...labeled cig. lighter...:) You may find some extra fuses in your glove box kit...if not, they are available at most Auto Parts stores...

Remove the fuse and look closely at the element in the fuse. You can tell by looking if it is burnt into. If you cannot and want to test it with an ohm meter, then set the meter to 200 ohms and touch the probes together to make sure the ohm meter is working properly. You should get a reading if it is. Now put the black probe on one connection of the fuse and the red probe on the other. If you get a reading the fuse is good. If not it is bad.

did you use a test light this would be the primary source for an accessory not to work unless the wires to the connector on the back side of the lighter itself has come undone or it may very well be the ground to it you need to visually inspect it a mirror may be of assistance to have a look if you cannot bend under neath the dash maby you can remove the ashtray to better view it

so you replaced a blown fuse to the cigar lighter and it still dosent work?.. then the wiring was compromised before the fuse blew..you need to repair bad wiring to cigar lighter and/or lighter itself internally melted /failed due to your air pump asking for more electricity than the cigar lighter could deliver..once repaired do not use the air pump there again ..good luck